I do not know what a "250 MG converter to 400 hps sun system light" means? First, you are going to want a ballast that is digital and can be turned down. I suggested the 400W digi because you can turn it down and people tend to go to bigger spaces rather than smaller ones. Also the higher wattage lights put out more lumens per watt than the smaller ones. For instance most 150W HPS put out 15,000 lumens or 100 lumens per watt. The 400W can put out 125 lumens per watt and the 600W 150 lumens per watt, so they do get more efficient as the wattage increases. You also want an air coolable hood. For a small space, a cool tube works great. I personally would not own a hood that was not air coolable, it makes that much difference. I am pretty sure that you will not be able to keep the space cool enough when you get to flowering and have as much light as you need.
I also do not know what McGee ducting and fan means? I doubt that you are going to find a carbon filter less than 4" and you are going to need a centrifuge type fan to pull the air through the filter--not all fans will work.
I understand living on a budget. I have to watch my pennies, too. I am trying to keep you from making buying mistakes that ultimately will end up costing you more money in the long run. More light means more bud. There is a cap to what the plants can utilize, but it is up somewheres around 7500-10,000 lumens per sq ft...maybe more. But unfortunately, there is a reason that cannabis costs as much as it does--it is not that easy or cheap to grow. You cannot plant acres of it like corn. Each plant needs individual care and they can be finicky. Adequate lighting and to a lesser degree, but still critical, proper ventilation are essential to a good grow. The biggest mistake that new growers make is under-lighting their spaces. It always looks like the lights put off so much light that it would just have to be enough, but cannabis is a light hog and loves lots of it in the correct spectrum.
You can get a real nice 400W setup on Amazon for about $150--come with a cool tube, hangers, MH and HPS bulbs. Very few white "closets" are made from materials that block light. Unless it is a closet designed as a darkroom or a grow tent, I doubt that it will block light 100%...and that is critical for flowering plants. Most of us that have been growing for any amount of time have a bunch of growing stuff that we tried and then went on to something better. We want to give you the benefit of our experience so you do not end up with a closet full of stuff that just isn't enough to do the job. Anytime I am contemplating buying something new, I still run it by the members here for their input.
You are a good daughter to help your mom out. I understand running around like a chicken with your head cut off, too. I was a single mother of 2, went to school full time and had a full time job. I know it is tough. But we cannot change the fact that growing cannabis takes a certain amount of money, time, energy, and love. We don't want to shoot you down or discourage you, but we do want you to know the way it is. Your 150W is enough to take you through vegging. You shouldn't need a carbon filter til flowering. Amazon also has a fan/filter combo for under $100, I believe. Save pennies, keep your eye open for good buys (check CL periodically, thrift stores, etc). Green MOJO for your plants.